The Multiplicity Conjecture (MC) of Huneke and Srinivasan provides upper and
lower bounds for the multiplicity of a Cohen-Macaulay algebra A in terms of
the shifts appearing in the modules of the minimal free resolution (MFR) of
A. All the examples studied so far have lead to conjecture (see [HZ] and
[MNR2]) that, moreover, the bounds of the MC are sharp if and only if A has
a pure MFR. Therefore, it seems a reasonable - and useful - idea to seek
better, if possibly {\it ad hoc}, bounds for particular classes of
Cohen-Macaulay algebras. In this work we will only consider the codimension 3
case. In the first part we will stick to the bounds of the MC, and show that
they hold for those algebras whose h-vector is that of a compressed algebra.
In the second part, we will (mainly) focus on the level case: we will construct
new conjectural upper and lower bounds for the multiplicity of a codimension 3
level algebra A, which can be expressed exclusively in terms of the
h-vector of A, and which are better than (or equal to) those provided by
the MC. Also, our bounds can be sharp even when the MFR of A is not pure.
Even though proving our bounds still appears too difficult a task in general,
we are already able to show them for some interesting classes of codimension 3
level algebras A: namely, when A is compressed, or when its h-vector
h(A) ends with (...,3,2). Also, we will prove our lower bound when h(A)
begins with (1,3,h2,...), where h2≤4, and our upper bound when h(A)
ends with (...,hc−1,hc), where hc−1≤hc+1.Comment: 22 pages. A few (non-substantial) changes. To appear in J. of Pure
and Appl. Algebr